Abstract
We propose an empirical Bayes method for high-dimensional linear regression models. Following an oracle approach that quantifies the error locally for each possible value of the parameter, we show that an empirical Bayes posterior contracts at the optimal rate at all parameters and leads to uniform size-optimal credible balls with guaranteed coverage under an "excessive bias restriction"condition. This condition gives rise to a new slicing of the entire space that is suitable for ensuring uniformity in uncertainty quantification. The obtained results immediately lead to optimal contraction and coverage properties for many conceivable classes simultaneously. The results are also extended to high-dimensional additive nonparametric regression models.
Original language | English |
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Pages (from-to) | 3113-3137 |
Number of pages | 25 |
Journal | Annals of Statistics |
Volume | 48 |
Issue number | 6 |
Early online date | 11 Dec 2020 |
DOIs | |
Publication status | Published - Dec 2020 |
Funding
PROOF OF COROLLARY 4. Consider the case pî = pˇi, the other case is similar. Note that θˆ has support Î. By part (ii) of Corollary 1, Î has cardinality |Î| ≤ τ0s(θ0) ≤ τ0s for some constantτ0>0withprobabilityatleast1−exp{−c′′s(θ0)log(ep/s(θ0))},wherec′′>0isa constant. Since (θˆ − θ0) is supported within an index set of cardinality at most (τ0 + 1)s(θ0), by the definition of the compatibility coefficient φ2, it follows that ‖X(θˆ − θ0)‖ ≥ φ2((τ0 + 1)s(θ0))‖X‖max‖θˆ − θ0‖. This leads to the first claim of the corollary. The proof for the ℓ1-case is similar. The last relation is inherited from Theorem 2. □ Acknowledgments. Research of the second author is partially supported by National Science Foundation (NSF) grant number DMS-1510238. A visit to VU Amsterdam was supported by visitor STAR grant from the Netherlands Organisation for Scientific Research (NWO).
Funders | Funder number |
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National Science Foundation | DMS-1510238 |
National Science Foundation | |
Nederlandse Organisatie voor Wetenschappelijk Onderzoek |
Keywords
- Coverage
- Credible ball
- Empirical Bayes
- Excessive bias restriction
- Oracle rate